INDUCTION MOTOR steady-state model (squirrel cage) MEP 1523 ELECTRIC DRIVES INDUCTION MOTOR steady-state model (squirrel cage)

Stator – 3-phase winding Rotor – squirrel cage / wound Construction 120o a c’ b’ c b a’ Stator – 3-phase winding Rotor – squirrel cage / wound

Construction Single N turn coil carrying current i Spans 180o elec Permeability of iron >> o → all MMF drop appear in airgap a  a’   /2 -/2 - Ni / 2 -Ni / 2

Construction Distributed winding – coils are distributed in several slots Nc for each slot  (3Nci)/2 (Nci)/2 - -/2 /2  

Distributed winding (full-pitch) Construction Distributed winding (full-pitch) The resultant MMF is the total contribution of MMF from each coil Considering only the space-fundamental component, Concentrated Distributed Distributed space fundamental Concentrated space fundamental

Phase a – sinusoidal distributed winding  Air–gap mmf F()   2

Sinusoidal winding for each phase produces space sinusoidal MMF and flux Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF Combination of 3 standing waves resulted in MMF wave rotating at: p – number of poles f – supply frequency

Rotating flux induced: emf in stator winding (known as back emf) Emf in rotor winding Rotor flux rotating at synchronous frequency Rotor current interact with flux producing torque Rotor ALWAYS rotate at frequency less than synchronous, i.e. at slip speed: sl = s – r Ratio between slip speed and synchronous speed known as slip

Stator voltage equation: Vs = Rs Is + j(2f)LlsIs + Eag Eag – airgap voltage or back emf Eag = k f ag Rotor voltage equation: Er = Rr Ir + js(2f)Llr Er – induced emf in rotor circuit Er /s = (Rr / s) Ir + j(2f)Llr

Per–phase equivalent circuit Llr Lls Ir Rs + Vs – + Eag – + Er/s – Is Lm Rr/s Im Rs – stator winding resistance Rr – rotor winding resistance Lls – stator leakage inductance Llr – rotor leakage inductance Lm – mutual inductance s – slip

We know Eg and Er related by Where a is the winding turn ratio The rotor parameters referred to stator are: rotor voltage equation becomes Eg = (Rr’ / s) Ir’ + j(2f)Llr’ Ir’

Per–phase equivalent circuit Rr’/s + Vs – Rs Lls Llr’ Eag Is Ir’ Im Lm Rs – stator winding resistance Rr’ – rotor winding resistance referred to stator Lls – stator leakage inductance Llr’ – rotor leakage inductance referred to stator Lm – mutual inductance Ir’ – rotor current referred to stator

Power and Torque Power is transferred from stator to rotor via air–gap, known as airgap power Lost in rotor winding Converted to mechanical power = (1–s)Pag  Pm = (1-s)Pag

Mechanical power, Pm = Tem r Power and Torque Mechanical power, Pm = Tem r But, ss = s - r  r = (1-s)s  Pag = Tem s Therefore torque is given by (based on approximate equivalent circuit):

Power and Torque sm Tem Pull out Torque (Tmax) Trated r 0 rated s s 1 0

Steady state performance The steady state performance can be calculated from equivalent circuit, e.g. using Matlab Rr’/s + Vs – Rs Lls Llr’ Eag Is Ir’ Im Lm

Steady state performance Rr’/s + Vs – Rs Lls Llr’ Eag Is Ir’ Im Lm e.g. 3–phase squirrel cage IM V = 460 V Rs= 0.25  Rr=0.2  Lr = Ls = 0.5/(2*pi*50) Lm=30/(2*pi*50) f = 50Hz p = 4

Steady state performance

Steady state performance